Define position by 3 Euler Angles and Radius
3 euler angles and a radius is not enough values to accomplish what you want. If you added an AXIS, than that may help, but then you also need the axis. If you say X axis, well that really means (1,0,0). After rotation with your euler angles will transform that (1,0,0) to something else. Then you can multiply by scaler distance to get the point you are looking for.

import mathutils
from math import radians

vec = mathutils.Vector(0.0, 0.0, 1.0))
eul = mathutils.Euler((0.0, math.radians(45.0), 0.0), 'XYZ').to_matrix()
scalar = 10.0

vec = vec * eul
position_of_object = vec * scalar